Lemma 4.18.4. Let $\mathcal{C}$ be a category. The following are equivalent:

Finite limits exist in $\mathcal{C}$.

Finite products and equalizers exist.

The category has a final object and fibre products exist.

Lemma 4.18.4. Let $\mathcal{C}$ be a category. The following are equivalent:

Finite limits exist in $\mathcal{C}$.

Finite products and equalizers exist.

The category has a final object and fibre products exist.

**Proof.**
Since finite products, fibre products, equalizers, and final objects are limits over finite index categories we see that (1) implies both (2) and (3). By Lemma 4.14.11 above we see that (2) implies (1). Assume (3). Note that the product $A \times B$ is the fibre product over the final object. If $a, b : A \to B$ are morphisms of $\mathcal{C}$, then the equalizer of $a, b$ is

\[ (A \times _{a, B, b} A)\times _{(\text{pr}_1, \text{pr}_2), A \times A, \Delta } A. \]

Thus (3) implies (2) and the lemma is proved. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: